arXiv subjects

Zhengfeng Ji

Publications and source records attributed to Zhengfeng Ji.

At least 19 recordsLinked to original sources

arXiv2026

Long-horizon autoformalization of a core theorem underlying MIP* = RE

Landmark mathematical formalizations have taken specialist teams years to complete. We present FormalFlow, a system that coordinates AI proving agents under human supervision to address statement drift and proof composition in long-horizon formalization. Drawing on software engineering principles and practices, it uses a shared blueprint to guide nested planning, proving and review loops. Agents strengthen verification and review throughout formalization. We completed a machine-checked Lean 4 proof of the quantum soundness of the classical low individual-degree test, a core theorem underlying MIP* = RE. Developing the proof took 63 days; greater parallelism could further reduce this time. The final library contains 126,367 lines of Lean code, all generated by agents. The formalization corrects side conditions and intermediate errors while preserving the published final error bound under corrected assumptions. This work provides a verified foundation for quantum complexity and demonstrates a route to affordable verification of major research proofs by small teams.

quant-ph↗

arXiv2026

DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians

We study the computational complexity of estimating the normalized trace $2^{-n}\mathrm{Tr}[f(A)]$ for a log-local Hamiltonian $A$ acting on $n$ qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions $f(x)$. We show that if $f(x)$ is a continuous function with approximate degree $Ω(\mathrm{poly}(n))$, then estimating $2^{-n}\mathrm{Tr}[f(A)]$ up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of $f(x)$. This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when $A$ is sparse, the classical query complexity of this problem is exponential in the approximate degree. Together, these results identify the approximate degree as the key parameter governing the complexity of normalized trace estimation: it characterizes both the quantum complexity (via efficient DQC1 algorithms) and the classical hardness, yielding an exponential quantum-classical separation. Our proof develops a unified framework that cleanly combines circuit-to-Hamiltonian constructions, periodic Jacobi operators, and tools from polynomial approximation theory, including the Chebyshev equioscillation theorem.

quant-ph↗

arXiv2026

Quantum Behaviors Are Not Semialgebraic

The conditional probabilities achievable by local measurements on a shared quantum state in a Bell scenario form the set of quantum behaviors, whose structure was studied by Tsirelson. In 1993, Tsirelson asked whether this set is semialgebraic, that is, describable by finite Boolean combinations of polynomial equations and inequalities. This question has remained open. We answer it in the negative: with four binary measurements per party, the set of finite-dimensional quantum behaviors, its closure, and the commuting-operator set are all nonsemialgebraic. More strongly, none admits a finite real-analytic description even locally near a particular classical behavior. These results rule out exact finite semidefinite representations and show that no finite level of the Navascués-Pironio-Acín hierarchy characterizes these sets exactly.

quant-ph↗

arXiv2026

Optimal Covariance Inflation under Gaussian Tilts

Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body $K\subseteq\mathbb{R}^n$, let $μ_{K,t} (\mathrm{d} x) \propto e^{-t\| x \| ^2} \mathbb{1}_K(x)\,\mathrm{d} x$, and let $Q_n$ be the supremum of $\|\operatorname{Cov}(μ_{K,t})\|_{\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=Θ(n^{2/5})$, closing the gap between the known $Ω(n^{1/3})$ lower bound and the $O(\sqrt{n\log(en)})$ upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance $Ω(n^{2/5})$.

math.PR↗

arXiv2026

Gate-Efficient Implementation of the Query-Optimal Time-Dependent Hamiltonian Simulation

The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

quant-ph↗

arXiv2026

Quantum Maximum Entropy Inference and Hamiltonian Learning

Maximum entropy inference and learning of graphical models are pivotal tasks in learning theory and optimization. This work extends algorithms for these problems, including generalized iterative scaling (GIS) and gradient descent (GD), to the quantum realm. While the generalization, known as quantum iterative scaling (QIS), is straightforward, the key challenge lies in the non-commutative nature of quantum problem instances, rendering the convergence rate analysis significantly more challenging than the classical case. Our principal technical contribution centers on a rigorous analysis of the convergence rates, involving the establishment of both lower and upper bounds on the spectral radius of the Jacobian matrix for each iteration of these algorithms. Furthermore, we explore quasi-Newton methods to enhance the performance of QIS and GD. Specifically, we propose using Anderson mixing and the L-BFGS method for QIS and GD, respectively. These quasi-Newton techniques exhibit remarkable efficiency gains, resulting in orders of magnitude improvements in performance. As an application, our algorithms provide a viable approach to designing Hamiltonian learning algorithms.

cs.LG↗

arXiv2026

Quantum Speedups for Log-Concave Sampling from Local Structure

For a convex function $f \colon \mathbb{R}^d \to \mathbb{R}$, the problem of sampling from a distribution proportional to $e^{-f(x)}$ is called log-concave sampling. In many practical scenarios, the function $f(x)$ turns out to admit a local decomposition $f(x) = \sum_{a=1}^R ψ_a(x_{S_a})$. In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause $ψ_a(\cdot)$, which can be computationally much cheaper than the queries to $f(x)$ itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using $\widetilde{O}(\sqrtκd)$ local queries, where $κ$ is the condition number. This improves the prior best classical result $\widetilde{O}(κd)$ due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result $\widetilde{O}(\sqrtκ d^2)$ implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.

quant-ph↗

arXiv2026

No-Go Theorems for Quantum Transport Metrics from Fixed Cost Operators

Coupling-based quantum optimal transport generalizes classical optimal transport by representing transport plans as bipartite states with prescribed marginals and evaluating their cost as the expectation of a fixed Hermitian operator. Friedland et al. [Phys. Rev. Lett. 129, 110402 (2022)] conjectured that the square root of the optimal cost associated with the SWAP projector is a metric in every dimension and that this property persists for nearby quantum cost matrices. Miller [arXiv:2607.07764] disproved both conjectures by constructing explicit diagonal qutrit counterexamples. Building on his analysis, we prove a uniform no-go theorem for standard couplings. In every dimension $d\geq3$, no fixed cost operator makes either the optimal cost or its square root a metric, with violations occurring already among commuting states. The obstruction persists under stabilization of the SWAP cost. For channel-induced couplings, global nonnegativity and vanishing self-cost force the cost operator to be zero, precluding point separation when $d\geq2$. Taken together, these no-go results show that fixed-cost coupling formulations do not lead to metrics on the full quantum state space.

quant-ph↗

arXiv2026

Quantum Nonlocality under Latency Constraints

Bell inequalities are bounds on the correlations between different parties obeying a local hidden variable theory. Here, "local" refers to spacetime locality: the parties cannot communicate their inputs because they must produce their outputs faster than the speed-of-light delay between them. In other words, the parties must satisfy a certain latency constraint. In this work, we explicitly incorporate spacetime locality into the formulation of Bell inequalities by imposing such a latency constraint. When the latency constraint is sufficiently tight such that no parties can communicate, this becomes a standard Bell scenario. When the latency constraint is relaxed such that a subset of the parties can communicate, we no longer have a Bell scenario, but we can again find a divide between classical and quantum behaviors. Hence, we observe that the classical-quantum gap should actually be a function of time. To study these more general scenarios, we introduce the mathematical framework of latency-constrained games, which models time-evolving input and output processes for spatially separated parties subject to finite communication speeds. This framework allows us to systematically study the weirdness of quantum mechanics in the "low-latency regime" where the speed-of-light delay is non-negligible. Latency-constrained games can describe real-time decision-making in real-world settings that are latency-sensitive, such as high-frequency trading and distributed systems, and can reveal the utility of quantum correlations in these settings.

quant-ph↗

arXiv2026

A Correlation-Gap Bound for Nonlinear Gaussian PCA

Principal component analysis (PCA) is optimal for the linear reconstruction of Gaussian data, a foundational property underlying its central role in algorithms and signal processing. Its nonlinear analogue, however, is notoriously subtle: in 2011, Mallat and Zeitouni conjectured that the Karhunen--Loève (KL) basis remains optimal even when the retained coordinates are chosen adaptively per sample, a property that would theoretically justify the ubiquitous pipeline of PCA followed by sparse thresholding. In this paper, we establish a $1+O(1/\sqrt{d})$-approximate version of the retained-energy form of the Mallat--Zeitouni conjecture, showing that the KL basis is within this factor of the optimal basis. This dimension-free comparison depends only on the number of retained coordinates and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows. It complements the universal-constant reconstruction-error comparison of Litvak and Tikhomirov (Ann. Appl. Probab., 2018), while providing a comparison naturally suited for algorithmic analysis. Our proof rests on a clean, conceptual reduction: we relax arbitrary rotations to a deterministic threshold bound via Schur--Horn majorization, and identify the remaining loss with the correlation gap of the rank-$d$ uniform matroid over Gaussian level sets.

cs.DS↗

arXiv2026

On the Complexity of the Circuit Width Problem

Montanaro's polynomial representation expresses amplitudes of quantum circuits over the gates $H$, $Z$, $CZ$, and $CCZ$ as normalized gaps of degree-three polynomials over $\mathbb{F}_2$. The normalization is governed by the circuit width $w(f)$, the minimum number of qubits in any circuit realizing a polynomial $f$. Thus, efficient width minimization would give an approximate-counting route toward a combinatorial characterization of $BQP$. We study the computational complexity of this parameter. For degree-three polynomials with no constant term, deciding whether $w(f)\le k$ is $NP$-complete, resolving Montanaro's open question. We also prove $NP$-hardness of approximation within any factor $49/48-ε$, and show via a twin-copy construction that the exact and approximation hardness results also hold for degree-two polynomials. Under the Exponential Time Hypothesis, the exact problem admits no $2^{o(n)}$-time algorithm when $k=Θ(n)$. Complementing these hardness results, we give a nondeterministic polynomial-time search algorithm using $2\log_2\binom{n}{k}=O(k\log(en/k))$ witness bits, and a constructive fixed-parameter algorithm parameterized by $k$ with running time $k^{6k+o(k)}n+O(m)$.

cs.CC↗

arXiv2026

Optimal Compilation of Syndrome Extraction Circuits for General Quantum LDPC Codes

Quantum error correcting codes (QECC) are essential for constructing large-scale quantum computers that deliver faithful results. As strong competitors to the conventional surface code, quantum low-density parity-check (qLDPC) codes are emerging rapidly: they offer high encoding rates while maintaining reasonable physical-qubit connectivity requirements. Despite the existence of numerous code constructions, a notable gap persists between these designs -- some of which remain purely theoretical -- and their circuit-level deployment. In this work, we propose Auto-Stabilizer-Check (ASC), a universal compilation framework that generates depth-optimal syndrome extraction circuits for arbitrary qLDPC codes. ASC leverages the sparsity of parity-check matrices and exploits the commutativity of X and Z stabilizer measurement subroutines to search for optimal compilation schemes. By iteratively invoking an SMT solver, ASC returns a depth-optimal solution if a satisfying assignment is found, and a near-optimal solution in cases of solver timeouts. Notably, ASC provides the first definitive answer to one of IBM's open problems: for all instances of bivariate bicycle (BB) code reported in their work, our compiler certifies that no depth-6 syndrome extraction circuit exists. Furthermore, by integrating ASC with an end-to-end evaluation framework -- one that assesses different compilation settings under a circuit-level noise model -- ASC reduces circuit depth by approximately 50% and achieves an average 7x-8x suppression of the logical error rate for general qLDPC codes, compared with as-soon-as-possible (ASAP) and coloration-based scheduling. ASC thus substantially reduces manual design overhead and demonstrates its strong potential to serve as a key component in accelerating hardware deployment of qLDPC codes.

quant-ph↗

arXiv2026

Lévy-Khintchine Structure Enables Fast-Forwardable Lindbladian Simulation

Simulation of open quantum systems is an area of active research in quantum algorithms. In this work, we revisit the connection between Markovian open-system dynamics and averages of Hamiltonian real-time evolutions, which we refer to as Hamiltonian twirling channels. By applying the Lévy-Khintchine representation theorem, we clarify when and how a dissipative dynamics can be realized using Hamiltonian twirling channels. Guided by the general theory, we explore Hamiltonian twirling with Gaussian, compound Poisson and symmetric stable distributions and their algorithmic implications. These give wide classes of Lindbladians that can be simulated in $Θ(t^{1/α})$ Hamiltonian simulation time without any extra ancilla or other quantum gates for $1\le α\le 2$. Moreover, we prove that these time complexities are asymptotically optimal using an information theoretic approach, which, to the best of our knowledge, is the first result of lower bounds on fast-forwarding simulation algorithms.

quant-ph↗

arXiv2026

Learning to Decode in Parallel: Self-Coordinating Neural Network for Real-Time Quantum Error Correction

Fast, reliable decoders are pivotal components for enabling fault-tolerant quantum computation (FTQC). Neural network decoders like AlphaQubit have demonstrated potential, achieving higher accuracy than traditional human-designed decoding algorithms. However, existing implementations of neural network decoders lack the parallelism required to decode the syndrome stream generated by a superconducting logical qubit in real time. Moreover, integrating AlphaQubit with sliding window-based parallel decoding schemes presents non-trivial challenges: AlphaQubit is trained solely to output a single bit corresponding to the global logical correction for an entire memory experiment, rather than local physical corrections that can be easily integrated. We address this issue by training a recurrent, transformer-based neural network specifically tailored for parallel window decoding. While it still outputs a single bit, we derive training labels from a consistent set of local corrections and train on various types of decoding windows simultaneously. This approach enables the network to self-coordinate across neighboring windows, facilitating high-accuracy parallel decoding of arbitrarily long memory experiments. As a result, we overcome the throughput bottleneck that previously precluded the use of AlphaQubit-type decoders in FTQC. Our work presents the first scalable, neural-network-based parallel decoding framework that simultaneously achieves SOTA accuracy and the stringent throughput required for real-time quantum error correction. Using an end-to-end experimental workflow, we benchmark our decoder on the Zuchongzhi 3.2 superconducting quantum processor on surface codes with distances up to 7, demonstrating its superior accuracy. Moreover, we demonstrate that, using our approach, a single TPU v6e is capable of decoding surface codes with distances up to 25 within 1us per decoding round.

quant-ph↗

arXiv2025

Scalable Quantum Error Mitigation with Neighbor-Informed Learning

Noise in quantum hardware is the primary obstacle to realizing the transformative potential of quantum computing. Quantum error mitigation (QEM) offers a promising pathway to enhance computational accuracy on near-term devices, yet existing methods face a difficult trade-off between performance, resource overhead, and theoretical guarantees. In this work, we introduce neighbor-informed learning (NIL), a versatile and scalable QEM framework that unifies and strengthens existing methods such as zero-noise extrapolation (ZNE) and probabilistic error cancellation (PEC), while offering improved flexibility, accuracy, efficiency, and robustness. NIL learns to predict the ideal output of a target quantum circuit from the noisy outputs of its structurally related ``neighbor'' circuits. A key innovation is our 2-design training method, which generates training data for our machine learning model. In contrast to conventional learning-based QEM protocols that create training circuits by replacing non-Clifford gates with uniformly random Clifford gates, our approach achieves higher accuracy and efficiency, as demonstrated by both theoretical analysis and numerical simulation. Furthermore, we prove that the required size of the training set scales only \emph{logarithmically} with the total number of neighbor circuits, enabling NIL to be applied to problems involving large-scale quantum circuits. Our work establishes a theoretically grounded and practically efficient framework for QEM, paving a viable path toward achieving quantum advantage on noisy hardware.

quant-ph↗

arXiv2025

Quantum Design Automation: Foundations, Challenges, and the Road Ahead

Quantum computing is transitioning from laboratory research to industrial deployment, yet significant challenges persist: system scalability and performance, fabrication yields, and the advancement of algorithms and applications. We emphasize that in building quantum computers -- spanning quantum chips, system integration, instruction sets, algorithms, and middleware such as quantum error correction schemes -- design is everywhere. In this paper, we advocate for a holistic design perspective in quantum computing, a perspective we argue is pivotal to unlocking innovative co-design opportunities and addressing the aforementioned key challenges. To equip readers with sufficient background for exploring co-optimization opportunities, we detail how interconnected computational methods and tools collaborate to enable end-to-end quantum computer design. This coverage encompasses critical stages -- such as chip layout design automation, high-fidelity system-level simulation, Hamiltonian derivation for quantum system modeling, control pulse simulation, decoherence analysis, and physical verification and testing -- followed by quantum instruction set design. We then proceed to quantum system and software development, including quantum circuit synthesis, quantum error correction and fault tolerance, and logic verification and testing. Through these discussions, we illustrate with concrete examples -- including co-optimizing quantum instruction sets with algorithmic considerations, customizing error correction circuits to hardware-specific constraints, and streamlining quantum chip design through tailored code design, among others. We hope that the detailed end-to-end design workflow as well as these examples will foster dialogue between the hardware and software communities, ultimately facilitating the translation of meaningful research findings into future quantum hardware implementations.

quant-ph↗

arXiv2025

LimTDD: A Compact Decision Diagram Integrating Tensor and Local Invertible Map Representations

Tensor networks serve as a powerful tool for efficiently representing and manipulating high-dimensional data in applications such as quantum physics, machine learning, and data compression. Tensor Decision Diagrams (TDDs) offer an efficient framework for tensor representation by leveraging decision diagram techniques. However, the current implementation of TDDs and other decision diagrams fail to exploit tensor isomorphisms, limiting their compression potential. This paper introduces Local Invertible Map Tensor Decision Diagrams (LimTDDs), an extension of TDDs that incorporates local invertible maps (LIMs) to achieve more compact representations. Unlike LIMDD, which uses Pauli operators for quantum states, LimTDD employs the $XP$-stabilizer group, enabling broader applicability across tensor-based tasks. We present efficient algorithms for normalization, slicing, addition, and contraction, critical for tensor network applications. Theoretical analysis demonstrates that LimTDDs achieve greater compactness than TDDs and, in best-case scenarios and for quantum state representations, offer exponential compression advantages over both TDDs and LIMDDs. Experimental results in quantum circuit tensor computation and simulation confirm LimTDD's superior efficiency. Open-source code is available at https://github.com/Veriqc/LimTDD.

cs.DS↗

arXiv2025

A Meta-Complexity Characterization of Minimal Quantum Cryptography

We give a meta-complexity characterization of EFI pairs, which are considered the "minimal" primitive in quantum cryptography (and are equivalent to quantum commitments). More precisely, we show that the existence of EFI pairs is equivalent to the following: there exists a non-uniformly samplable distribution over pure states such that the problem of estimating a certain Kolmogorov-like complexity measure is hard given a single copy. A key technical step in our proof, which may be of independent interest, is to show that the existence of EFI pairs is equivalent to the existence of non-uniform single-copy secure pseudorandom state generators (nu 1-PRS). As a corollary, we get an alternative, arguably simpler, construction of a universal EFI pair.

quant-ph↗